29. Simple Interest versus Compound Interest. First Simple Bank pays 7 percent simple interest on its investment accounts. If First Complex Bank pays interest on its accounts compounded annually, what rate should the bank set if it wants to match First Simple Bunk over an investment horizon of 10 years?
30. Calculating Annuities Due. You want to buy a new sports car from Muscle Motors for $48,000. The contract is in the form of a 60-month annuity due at a 7.45 percent APR. What will your monthly payment be?
31. Calculating Interest Expense. You receive a credit card application from Shady Banks Savings and Loan offering an introductory rate of l .5 percent per year, compounded monthly for the first six months, increasing thereafter to 18 percent compounded monthly. Assuming you transfer the $10,000 balance from your existing credit card and make no subsequent payments, how much interest will you owe at the end of the first year?
32. Calculating the Number of Periods. You are saving to buy a $175,000 house. There are two competing banks in your area, both offering certificates of deposit yielding 5 percent. How long will it take your initial $89,000 investment to reach the desired level at First Bank, which pays simple interest? How long at Second Bank, which compounds interest monthly?
33. Calculating Future Values. You have an investment that will pay you 1.05 percent per month. How much will you have per dollar invested in one year? In two years?
34. Calculating Annuity Interest Rates. Although you may know William Shakespeare from his classic literature, what is not well-known is that he was an astute investor. In 1604. when he was 40 and writing King Lear, Shakespeare grew worried about his eventual retirement. Afraid that he would become like King Lear in his retirement and beg hospitality from his children, be purchased grain "tithes," or shares in farm output, for 440 pounds. The tithes paid him 60 pounds per year for 31 years. Even though he died at the age of 52, his children received the remaining payments. What interest rate did the Bard of Avon receive on this investment?
35. Comparing Cash Flow Streams. You've just joined the investment banking firm of Dewey, Cheatum, and Howe. They've offered you two different salary arrangements. You can have $7,500 per month for the next two years, or you can have $6,000 per month for the next two years, along with a $30,000 signing bonus today. If the interest rate is 7 percent compounded monthly, which do you prefer?
36. Calculating Present Value of Annuities. Peter Lynchpin wants to sell you an investment contract that pays equal SI5,000 amounts at the end of each of the next 20 years. If you require an effective annual return of 13 percent on this investment, how much will you pay for the contract today?
37. Calculating Rates of Return. You're trying to choose between two different investments, both of which have up-front costs of $75,000. Investment G returns $120,000 in six years. Investment H returns $220,000 in 13 years. Which of these investments has the higher return?
38. Present Value and Interest Rates. What is the relationship between the value of an annuity and the level of interest rates? Suppose you just bought a 10-year annuity of $10,000 per year at the current interest rate of 10 percent per year. What happens to the value of your investment if interest rates suddenly drop to 5 percent? What if interest rates suddenly rise to 15 percent?
39. Calculating the Number of Payments. You're prepared to make monthly payments of $250, beginning at the end of this month, into an account that pays
10 percent interest compounded monthly. How many payments will you have made when your account balance reaches $50,000?
40. Calculating Annuity Present Values. You want to borrow $75,000 from your local bank to buy a new sailboat. You can afford to make monthly payments of $1,600. hut no more. Assuming, monthly compounding, what is the highest rate you can afford on a 60-month APR loan?
41. Calculating Present Values. In 2001, Alex Rodriguez signed the biggest contract in sports history, reported to be worth $252 million. Including a deferred signing bonus, he would receive $24 million per year from 2001.-2004, $27 million in 2005, $25 million in 2006, and $26 million per year from 2007-2010. If the appropriate interest rate is 11 percent, what kind of deal did the inrielder snag? Assume the first payment is made immediately, and the next payment occurs in exactly one year.
42. Calculating Present Values. 2001 was a very good salary year for baseball players. In addition to Alex Rodriguez's contract. Manny Ramirez signed an 8-year contract reported to be worth SJ60 million. Including a deferred signing bonus, he would receive $16 million in 2001, $18.5 million in 2002. $21 million in 2003, $23.5 million in 2004, $23 million in 2005, $20 million in 2006. $ 1S million in 2007. and $20 million in 2008. If the appropriate interest rate is 11 percent, what kind of deal did the outfielder catch? Assume the first payment is made immediately, and the next payment occurs in cxactly one year.
43. EAR versus APR. You have just purchased a new warehouse. To finance the purchase, you've arranged for a 30-year mortgage loan for 80 percent of the $2,500,000 purchase price. The monthly payment on this loan will be $13,400. What is the APR on this loan? The EAR?
44. Annuity Values. You are planning your retirement in 10 years. You currently have $150,000 in a bond account and $450,000 in a stock account. You plan to add $9,000 per year at the end of each of the next 10 years to your bond account. The stock account will earn an 11.5 percent return and the bond account will earn a
7.5 percent return. When you retire, you plan to withdraw an equal amount for each of the next 25 years at the end of each year and have nothing left. Additionally, when you relire you will transfer your money to an account that earns 6.75 percent. How much can you withdraw each year?
45. Calculating Annuities Due. You want to buy a new sports car from Muscle Motors for $6! .000. The contract is in the form of a 60-month annuity due at an 8.15 percent APR. What will your monthly payment be?
46. Calculating Annuities Due. Suppose you are going to receive SI0,000 per year for five years. The appropriate interest rate is 11 percent.
a. What is the present value of the payments if they are in the form of an ordinary annuity? What is the present value if the payments are an annuity due?
b. Suppose you plan to invest the payments for five years. What is the future value if the payments are an ordinary annuity? What if the payments are an annuity due?
e. Which has the highest present value, the ordinary annuity or annuity due? Which has the highest future value? Will this always be true?
47. Annuity and Perpetuity Values. Mary is going to receive a 30-year annuity of $8,000. Nancy is going to receive a perpetuity of $8,000. If the appropriate interest rate is 9 percent, how much more is Nancy's cash flow worth?
48. Calculating Present Values. A 5-year annuity of 10 $9,000 semiannual payments will begin 9 years from now. with Lhe first payment coming 9.5 years from now. If the discount rate is 11 perccnt compounded semiannually, what is the value of this annuity five years from now? What is the value three years from now? What is the current value of the annuity?
49. Present Value and Multiple Cash Flows. What is the present value of SI.450 per y ear, at a discount rate of 9 percent, if the first payment is received 6 years from now and the last payment is received 20 years from now?
50. Variable Interest Rates. A 10-year annuity pays $2,500 per month, and payments are made at (he end of each month. If the interest rate is 11 percent compounded monthly for the first four years, and 8 percent compounded monthly thereafter, what is the present value of the annuity?
51. Comparing Cash Flow Streams. You have your choice of two investment accounts. Investment A is a 10-year annuity that features end-of-month $1,200 payments and has an interest rate of 7 percent compounded monthly. Investment
B is a 9 percent annually compounded lump-sum investment, also good for 10 years. How much money would you need to invest in B today for it to be worth as much as Investment A 10 years from now?
52. Calculating Present Value of a Perpetuity. Given an interest rate of 5.45 percent per year, what is the value at year t — 7 of a perpetual stream of $2,500 payments that begin at year t = 20?
53. Calculating EAR. A local finance company quotes a 15 percent interest rate on one-year loans. So, if you borrow $20,000, the interest for the year will be $3,000. Because you must repay a total of $23,000 in one year, the finance company requires you to pay $23,000/12, or $1,916.67, per month over the next 12 months. Is this a
15 percent loan? What rate would legally have to be quoted? What is the effective annual rate?
54. Calculating Future Values. If today is Year 0, what is the future value of the following cash flows five years from now? What is the future value 10 years from now? Assume an interest rate of 9.4 percent per year.
55. Amortization with Equal Payments. Prepare an amortization schedule for a three-year loan of $75,000. The interest rate is 9 percent per year, and the loan calls for equal annual payments. How much interest is paid in the third year? How much total interest is paid over the life of the loan?
56. Amortization with Equal Principal Payments. Rework Problem 55 assuming that the loan agreement calls for a principal reduction of $25,000 every year instead of equal annual payments.
57. Discount Interest Loans. This question illustrates what is known as discount interest. Imagine you are discussing a loan with a somewhat unscrupulous lender. You want to borrow $12,000 for one year. The interest rate is 13 percent. You and the lender agree that the interest on the loan will be .13 X $12,000 = $1,560. So the lender deducts this interest amount from the loan up front and gives you $10,440. In this case, we say that the discount is $1,560. What's wrong here?
58. Calculating Annuity Values. You are serving on a jury. A plaintiff is suing the city for injuries sustained after a freak street sweeper accident. In the trial, doctors testified that it will be five years before the plaintiff is able to return to work. The jury has already decided in favor of the plaintiff. You are the foreperson of the jury and propose that the jury give the plaintiff an award to cover the following: (a) The present value of two years' back pay. The plaintiff's annual salary for the last two years would have been $44,000 and $46,000, respectively, (b) The present value of five years' future salary. You assume the salary will be $49,000 per year, (c) $100,000 for pain and suffering, (d) $20,000 for court costs. Assume that the salary payments are equal amounts paid at the
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